CALCULATE THE TOP QUARK MASS
TEACHER NOTES
DESCRIPTION
In 1995 physicists at Fermilab announced the discovery of a new particle dubbed “Top
Quark.” The discovery of this particle completed the Standard Model chart until
discovery of the Higgs boson in 2012. Despite the use of gigantic detectors using
groundbreaking technology, the data analysis involved concepts that are a part of the
standard curriculum for high school physics.
In this activity, students use momentum conservation, energy conservation and twodimensional vector addition to calculate the mass of the heaviest of the six known quarks.
They gather data from data plots from Fermilab’s DØ experiment. The events were
chosen carefully; all of the decay products moved in a plane perpendicular to the beam.
This allows for analysis to take place in two dimensions instead of three. The use of twodimensional vector analysis fits in nicely with a conservation laws unit or as a practice
problem for the vector analysis unit.
STANDARDS ADDRESSED
Next Generation Science Standards
Science and Engineering Practices
4. Analyzing and interpreting data
5. Using mathematics and analytical thinking
6. Constructing explanations
7. Engaging in arguments from evidence
Crosscutting Concepts
1. Observed patterns
4. Systems and system models
5. Energy and matter
7. Stability and change
Common Core Literacy Standards
Reading
9-12.3 Follow precisely a complex multistep procedure . . .
9-12.4 Determine the meaning of symbols, key terms . . .
9-12.7 Translate quantitative or technical information . . .
Common Core Mathematics Standards
MP2. Reason abstractly and quantitatively.
MP4. Model with mathematics.
MP5. Use appropriate tools strategically.
MP6. Attend to precision.
ENDURING UNDERSTANDINGS
Particle physicists use conservation of energy and momentum to discovery the mass of
fundamental particles.
LEARNING OBJECTIVES
Students will know and be able to:
• Use conservation of momentum and energy to determine the mass of a top quark.
• Explain the importance of identifying the missing momentum carried away from
the event by the neutrino.
• Describe the properties of a neutrino that make it impossible to detect in the DØ
detector.
• Explain the importance of considering the results of several experiments before
announcing discoveries.
PRIOR KNOWLEDGE
Students must be able to add vectors in two dimensions and be able to use energy and
momentum units common to particle physics. (Momentum–eV/c, energy–eV/c2)
BACKGROUND MATERIAL
Useful links to describe how detectors work:
https://home.cern/about/how-detector-works
http://lutece.fnal.gov/Papers/PhysNews95.html
RESOURCES/MATERIALS
Students will need a ruler, a protractor, calculator or spreadsheet and data from the
following link: http://ed.fnal.gov/samplers/hsphys/activities/thumbnails_pdf.html.
IMPLEMENTATION
This activity was developed by physics teacher Bob Grimm. Students use printed event
images, ruler and protractor to analyze the data. This activity requires averaging as many
as ten data points per class to yield a more accurate result.
The students analyze DØ top-quark event images, one real and three Monte Carlo, chosen
because the events happened in the plane perpendicular to the beam line. This means that
students will get good results using two-dimensional vector analysis. Students use a ruler
and protractor to add particle momentum vectors, discover the magnitude of the missing
neutrino momentum, and calculate the mass of the top quark.
When Bob used this activity, his students were studying conservation laws, so the activity
fit right in. They didn't know about quarks, etc., but they spent some time, before the lab
day, looking at images taken from the website that describe the top quark experiment. So,
with their particle questions addressed, they were able to focus on the vector addition
portion of the activity. This approach is a good use of instructional time; on lab day, the
students' questions about vectors and momentum were not confused by questions about
particles. This is not to say they did not have questions about particles. Indeed, the
questions they had were fairly sophisticated. But they did not seem distracted by details
of the experiment.
Bob recommends keeping in mind the following "I" ideas while leading your students
through this activity:
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•
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Invest time in describing the experiment.
Ignore errors in the direction of the neutrino momentum vector.
Integrate your students' results. Averaging a large set of data is critical.
Indistinguishable units. Near the speed of light, mass = momentum = energy.
The key to finding the momentum carried away by the neutrino is to determine the
“missing Pt” (transverse momentum). Since the detector can’t see neutrinos—they barely
interact with matter—the students have to look at all of the momentum recorded in the
event and then apply momentum conservation to determine what is needed to make the
system’s net momentum zero. Recall that energy and momentum are equal at these
energies. If you have never done this before, the process is:
• Use a protractor to find the angle θ that the lines through the centers of all jets and
the muon tracks make with the x-axis.
• The magnitude of the momentum p for all the jets and muons is given on the
plot. Find px = p cos(θ) and py = p sin(θ) for all jets and muons.
• Find px total_observed and py total_observed.
• Recall that the center of mass momentum before the collision is zero; therefore,
since momentum is conserved, the vector sum of momenta after the collision must
also be zero. Since we can account for all of the momentum except the missing
neutrino momentum, the x and y component of the neutrino momentum are
pneutrino x = -px total_observed and pneutrino y = -py total_observed.
• The magnitude of the neutrino momentum is pneutrino = (pneutrino x2 + pneutrino y2)1/2.
• Add up the energies of all jets, muons, and the neutrino to find the rest energy of
the top-antitop pair.
• Since the event is a top-antitop event, total energy divided by two results in the
mass of the top quark. (in units of GeV/c2)
ASSESSMENT
Consider asking the students questions such as:
Can you explain the mathematical model for finding the missing momentum carried off
by the neutrino?
• Choose a coordinate system.
• Measure the angle of all vectors relative to the chosen x-axis.
• Correctly determine the x-component and y-component of each momentum vector.
• Find the sum of the x-components and y-components.
• Indicating that the vector components should add to zero, determine the xcomponent momentum and y-component momentum of the neutrino that is needed
to make the components’ sums equal to zero.
• Use the neutrino x-component and y-component to determine the magnitude of the
missing neutrino momentum.
Determine the energy of the neutrino must be the same as the magnitude of the
momentum of the neutrino when appropriate units are chosen.
• Start with Einstein’s equation E2 = p2c2 + (mc2)2.
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In the correct units, the equation reduces to E2 = p2 + m2.
The mass of the neutrino is negligible at these energy levels, so E = p.
Explain how conservation of energy is used to determine the mass of the top quark.
• The sum of the energy of the jets, muons, and neutrino must equal the mass of the
top quark/top antiquark pair.
Describe the properties of a neutrino that make it impossible to detect in the DØ detector.
• The neutrino has no charge and therefore does not interact with the tracking
section of the detector or the electromagnetic calorimeter.
• The neutrino has such small mass, it does not interact with matter and therefore
will not be detectable in the hadron calorimeter or the muon detector sections of
the detector.
Compare your individual result with the value determined from the class average.
• Were there outliers?
• How close was the class average to the value determined by scientists?
• Was the value determined by scientists within the range of the values found by
your class?
Use these results to describe why scientists want repeatability of results before
announcing discoveries.